Control systems operate under the assumption that sensors are trustworthy. Yet, when communication channels are unprotected or sensors are accessible from networked stations, malicious users can compromise the system by spoofing the measured information. We consider a linear time-invariant system with a single sensor, where the state is estimated by a Kalman filter. We assume the presence of an attacker with the ability to modify the measurements arbitrarily, which are then processed by the Kalman filter for as long as the attacker remains undetected. The objective of the attacker is to maximize the mean square error of the Kalman filter. We adopt a notion of attack stealthiness based on the Kullback-Leibler divergence measure, and characterize the worst case degradation induced by an attacker with a fixed stealthiness level. Additionally, we characterize optimal attack strategies that achieve our bound of performance degradation, thereby proving tightness of our result.